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Integration is one of the most interesting topics in Mathematics. It has a wide area of applications in very different aspects of engineering and science. There are numerous ways to tackle integration problems. For elementary ones please refer to the following

For the most part, I don't want you to be afraid by the word ''advanced'' the most important factor is practice and practice as long as you have basic knowledge of elementary integral calculus you should not get afraid. Some exercises here will require basic knowledge of certain properties of special functions which I will introduce before tackling certain problems (I will not focus on the proof) .Furthermore , basic background of complex variables will be such a great help here. I will try to leave certain problems for the reader with a final answer to try in your leisure time .

Finally , I will try to post an exercise every day so if you find any mistake don't hesitate to inform me . Furthermore , if you have any comments , face any problem understanding something or you want to show me your work just send me a pm or post it

This is one of the most commonly used techniques to solve numerous numbers of questions I will be using this technique to solve many other exercises in the coming posts

Assume that we have the following function of two variables :

$ \displaystyle \int ^{b}_a \, f(x,y) \, dx$

Then we can differentiate it with respect to y provided that f has partial continuous derivative on a chosen interval.

$ \displaystyle F'(y) = \int^{b}_{a} f_{y}(x,y)\,dx$

Now using this in many problems is not that clear you have to think a lot to get the required answer because many integral questions are just in one variable so you add the second variable and assume it is a function of two variables .

Assume we want to solve the following integral :

$ \displaystyle \int ^{1}_{0} \, \frac{x^2-1}{\ln (x) }$

Now that seems very difficult to solve but using this technique we can solve it easily no matter how much power is x raised to . So the crux move is to decide ,where to put the second variable ! So the problem with the integral is that we have a logarithm in the denominator which makes the problem so difficult to tackle !

I already gave this as a practice problem I will solve it now to check your solution :

We are given the following :

$$ \int^{\infty}_0 \frac{\sin(x) }{x}\, dx $$

This problem can be solved by many ways , but here we will try to solve it by differentiation .

So as I described earlier in the previous examples it is generally not so easy to find the function with two variables .

Actually this step might require trial and error techniques until we get the desired result , so don't just give up if an approach merely doesn't work !.

$ \displaystyle F(a)=\int^{\infty}_0 \frac{\sin(ax) }{x}\, dx$

Let us try this one :

If we differentiated with respect to a we get the following :

$ \displaystyle F'(a)=\int^{\infty}_0 \cos(ax) \, dx$

But unfortunately this integral doesn't converge , so this is not the correct one .

Well, that seemed hopeless , but you should benefit from mistakes . The previous integral will converge if there is an exponential (This is merely the Laplace transform which I will illustrate later ... ).

So let us try the following :

$ \displaystyle F(a)=\int^{\infty}_0 \frac{\sin(x) e^{-ax}}{x}\, dx$

Take the derivative to get :

$ \displaystyle F'(a)=-\int^{\infty}_0 \sin(x) e^{-ax}\, dx$

Now this is easy to solve we can use integration by parts twice to get the following :

As we have seen in the previous post hyperbolic functions can be so helpful as they help us simplify a lot of operations in a way that is really neat. Also, they help us handle complex variables to get a result that is completely free of complex numbers. This is basically because of the ability to convert from trigonometric functions though Euler formula.

Working with hyperbolic functions for the most part requires a lot of practice. It might not be so clear that the function can be integrated , this will help us in complex integration in the future .

Laplace transform is a very powerful transform. It can be used in many applications . For example, it can be used to solve Differential Equations and its rules can be used to solve integration problems .

Note : Just as a basic introduction to special functions, you can think of gamma function as the following : $n!=\Gamma{(n+1)}$ we can extend n to exist in the whole complex plane {except n is a negative integer }.

The beta function is so interesting , we will explain some of its basic properties later .It can be used to solve many integrals.
For the proof :

This is a very powerful rule it is basically saying that if we have a function divided by its independent variable and we integrated it in the half positive plane then we can transform it into an integral wrt to the Laplace transform of the function .

It is an important thing to get used to the symbol $\Gamma$. I am sure that you are saying (boring...) that this seems elementary , but my main aim here is to let you practice the new symbol and get used to solving some problems using it .